This e-book constitutes the completely refereed post-conference court cases of the thirteenth foreign convention on digital structures and Multimedia, VSMM 2007, held in Brisbane, Australia, in September 2007. The 18 revised complete papers provided have been conscientiously reviewed and chosen from ninety seven preliminary submissions in the course of rounds of reviewing and development.

Elkies To do this we must be able to compute numerically the rational function t : ∼ H/Γ ∗(1)→P1 . Equivalently, we need to associate to each t ∈ P1 a representative of its corresponding Γ ∗(1)-orbit in H. We noted already that this is done, up to a fractional linear transformation over C, by the quotient of two hypergeometric functions in t. To fix the transformation we need images of three points, and we naturally choose the elliptic points t = 0, 1, ∞. These go to fixed points of s2 , s4 , s6 ∈ Γ ∗(1), and to find those fixed points we need an explicit action of Γ ∗(1) on H.

This determines the cover up to K-isomorphism the curve X0 ((τ )) has genus 0, and we can choose coordinates x on that curve and t on X (1) such that t(P3 ) = ∞ and t = x3 − 3cx for some c = 0 — but not the location of the unramified point P2 relative to the other three elliptic points. To determine that we once again use the involution, this time w(τ) , of X0 ((τ )): this involution fixes the point above P2 corresponding to its self-isogeny, and pairs the other two preimages of P2 with the simple preimages of P2 , P2 .

Even in the simplest case Σ = {2, 3} where Γ ∗(1) is a triangle group and all the covers X0∗(l)/X ∗ (1) are in principle determined by their ramifications, finding those covers seems at present a difficult problem once l gets much larger than the few primes we have dealt with here. This is the case even when l is still small enough that X0∗ (l) has genus small enough, say g ≤ 5, that the curve should have a simple model in projective space. For instance, according to 35 the curve X0∗ (73) has genus 1.